Rip secret door, showing no trace of edges, into shade smooth sphere
As @Hikariztw has suggested, you need to give the cut sphere some custom normals.


On the cut sphere, check 'Auto Smooth' in its Data tab > 'Normals' panel, to enable vertex-per-face normals
Retain an intact copy of the sphere, in exactly the same place as the cut one. You can M,move that to another collection, and hide it, to avoid confusion. I've called mine 'Form'.
Give the cut sphere a Data Transfer modifier, settings as shown, to copy the normals from the intact sphere to the cut one.




You can apply the modifier, and dispose of the intact sphere to tidy up, if you like.

EDIT: in 2.8, you can evaluate the normals in Object space by unchecking the little collection icon at the right of the 'Source Object' field. That way, the target 'Form' does not have to be in exactly the same location or orientation as the cut sphere.. you can just shove it off to one side.